Analysis of Intermolecular Interaction among Pectin Molecules in Aqueous Sugar

Solutions

Yukinori Sato1* and Osato miyawaKi

2

1 Laboratory of Food Science and Technology, Kochi Women's University, Eikokuji 5-15, Kochi 780-8515, Japan2 Department of Food Science, Ishikawa Prefectural University, Suematsu 1-308, Nonoichi, Ishikawa 921-8836, Japan

Received April 26, 2007; Accepted January 28, 2008

Although the viscosity of aqueous solution of high methoxyl pectin is known to increase drastically when sugars coexist, the detailed mechanism for the increase in viscosity was not fully understood. There-fore, the viscosity of citrus and apple pectin solutions with various sugars compositions was measured with temperature varying from 5 to 40℃ to analyze the intermolecular interactions among pectin mol-ecules. For single-composition pectin solutions, the activation energy for viscosity, Ea, increased from 17.5 to 31.9 kJ/mol with an increase in pectin concentration up to 2% for the case of citrus pectin, reflecting the increase in pectin-pectin interaction. For pectin solutions with coexisting sugars, Ea increased more with increasing sugar concentration. When compared at the same water activity, the increase in Ea is also dependent on the type of sugar. Sugars with stronger solvent-ordering activity produced greater increases in Ea. These results suggest that sugars increase the pectin-pectin interaction both through their own hy-dration effect, which enhances the hydrogen bonding among pectin molecules, and through the solvent-ordering effect to enhance the hydrophobic interaction.

Keywords: pectin, viscosity, activation energy, hydrogen bonding, hydrophobic interaction

*To whom correspondence should be addressed.

Email: yukisato@cc.kochi-wu.ac.jp

IntroductionPectin is a hydrocolloid predominantly comprising ga-

lacturonic acid and its methyl ester form, and it has been

frequently utilized in food applications including as a gelling

agent and thickener (Thakur et al., 1997). Physical properties

of pectin solutions depend on the molecular size of pectin

molecules (Ishihara, 1992), the degree of methoxylation (Pip-

pen et al., 1953; Oakenfull and Scott, 1984), the degree of

dissociation of carboxylic groups (Michel et al., 1982), sub-

stituents of the carboxyl groups (Axelos and Thibault, 1991),

and the rhamnose content in the pectin molecule (Axelos and

Thibault, 1991). In addition, the solvent properties character-

ized by pH, ionic strength, and the solvent-ordering also play

important roles in determining the physical properties of pec-

tin solutions (Pippen et al., 1953; Oakenfull and Scott, 1984;

Chen and Joslyn, 1967; Sato et al., 2004).

When methoxyl content is low, the solution property of

pectin is dependent on the ionic linkages via carboxyl groups

belonging to different chains to form a structure resembling

an egg-box (Thakur et al., 1997). When methoxyl content

is high, the solution properties of pectin are known to be

strongly affected by the coexistence of sugars (Chen and

Joslyn, 1967; Crandall and Wicker, 1986; Kar and Arslan,

1999a; Bulone et al., 2002; Fishman et al., 2004; Sato et al.,

2004). However, the detailed mechanism of the effect of sug-

ars has been not fully understood.

In this paper, the viscosity of pectin solutions with vari-

ous sugars is systematically measured by changing sugar

concentration and temperature. The activation energy for

viscosity was obtained at various conditions to analyze the

intermolecular interactions among pectin molecules.

Materials and MethodsMaterials Glucose and sucrose were obtained from Na-

calai Tesque, Co. (Kyoto, Japan), mannose and maltose from

Wako Chemical Industries (Osaka, Japan), trehalose from

Tokyo Kasei Organic Chemicals (Tokyo, Japan), and ribose

from Sigma (St. Louis, MO, USA). For all the sugars, the

water content was analyzed by drying in oven at 110-120℃

Food Sci. Technol. Res., 14 (3), 232–238, 2008

233

to accurately calculate the sugar concentration.

Pectins from citrus and apple fruits were purchased from

Sigma. According to the technical data attached, the galact-

uronic acid and metoxyl content for the citrus pectin were

79.5% and 8.1% (DM, degree of esterification = 57.9%),

respectively. For the apple pectin, these were measured to

be 79.0% and 8.7% (DM=62.1%), respectively (Sato et al.,

2004). These pectins are classified as high-methoxyl pectins

(Glenn, 1953). All the other reagents used were of reagent

grade and were used without further purification.

Measurement of water activity For sugar solutions, wa-

ter activity (Aw) is expected to be described by the following

equation (Kozak et al., 1968):

Aw = (1 – Xs) exp (αXs2) (1)

where XS is the molar fraction of solute and the parameter α

is an experimentally determined constant, which represents

the aqueous solvent-ordering (Miyawaki et al., 1997). The

parameter α for sugars are -1.699, -1.929, -2.734, -7.405,

-8.775, and -9.549 for ribose, mannose, glucose, sucrose,

trehalose, and maltose, respectively (Miyawaki et al., 1997;

Sato et al., 2004).

When pectin coexisted with sugar, the effect of pectin

coexistence on Aw was neglected because of the much lower

molar concentration of pectin compared with sugars so that

the AW value without pectin was employed. Average molecu-

lar weight of pectin is reported to be over 105 daltons using

high-performance size exclusion chromatography (Howard,

1980). However, when the determining the precise effect of

the coexistence of pectin was necessary, AW was measured

by a water activity meter (Aqua Lab CX-3, AINEX, Tokyo,

Japan) at 25℃.

Viscosity measurement for pectin solutions Pectin pow-

der was suspended and dissolved to be 2% in distilled water

containing various concentration of sugars at 50℃. The

sample was kept overnight at 50℃ until use. Then, 7 mL of

the sample solution was poured into the sample cell, and the

apparent viscosity was measured by a rotational viscometer

(B8L, Tokimec, Tokyo, Japan) with rotational speed varying

from 0.5 to 100 rpm. Because of the shear-thinning property

of the pectin solution sample (Marcotte et al., 2001), the ap-

parent viscosity decreased with an increase in the rotational

speed, then stabilized in the rotational speed range typically

higher than 30 rpm (apparent shear rate=39.69 s-1).

For sugar solutions without pectin, the dynamic viscos-

ity was measured by Cannon-Fenske viscometer (Shibata,

Tokyo, Japan) since Newtonian flow is expected in this case.

The viscosity was calculated from the dynamic viscosity and

the density measured by a specific gravity meter (DA-130,

Kyoto Electronics, Kyoto, Japan).

Results and DiscussionActivation energy for viscosity of sugar solutions Figure

1 shows the effect of AW on the viscosity of sugar solutions

at 25℃, ηL, which could be described well by the following

2nd order polynomial of Aw.

ηL = ηW + a (1 – AW) + b (1 – AW)2 (2)

where ηW is the viscosity of pure water, a represents the

Einstein’s volumetric effect of solute and its hydration (Her-

skovits and Kelly, 1973), and b represents the nonlinear term.

Table 1 lists the experimental constants a and b along

with ηW for various sugar solutions at various temperatures,

which show good applicability of Eq. (2) to all the sugar

solutions tested here. From ηW , a, and b in Table 1, the tem-

perature dependence of ηL could be calculated at various

temperatures at fixed Aw, which is expected to be described

by an Arrhenius-type equation (Scott Blair, 1953; Kar and

Arslan, 1999b) as follows:

ηL = η0 exp(Ea/RT) (3)

where η0 is a pre-exponential factor (mPa·s), Ea is the ac-

tivation energy for viscosity (kJ/mol), R is the gas constant

(=8.3145 J/mol/K), and T is the absolute temperature (K).

According to Eq. (3), the logarithm of ηL showed a strong

linear relationship to the inverse of absolute temperature

(1/T ), as shown in Figure 2. From the slope of Figure 2, Ea

was calculated. Results are listed in Table 2 for various sugar

solutions at various Aw. The Ea slightly increased with a de-

crease in Aw and was higher for solutions with disaccharides

than those with monosaccharides.

Intermolecular Interaction of Pectin

0

0.5

1

1.5

2

2.5

0.980.9850.990.9951

SucroseGlucoseRibose

ηL(mPa・

s)

Aw

Fig. 1. Viscosity of sugar solutions at 25℃.

234 Y. Sato et al.

Sugar Temperature (℃) ηW** a b R

Ribose 5 1.521 26.56 835.15 0.997

10 1.307 15.78 849.46 0.997

15 1.135 10.17 1037.70 0.994

20 1.002 12.52 512.99 0.997

25 0.890 13.50 243.20 0.997

30 0.797 10.77 256.19 0.999

35 0.719 10.97 147.93 1.000

40 0.653 9.41 176.23 1.000

Mannose 5 41.80 1012.90 0.997

10 30.66 923.12 0.998

15 23.99 850.67 0.999

20 19.41 731.51 0.998

25 20.60 288.50 0.999

30 17.29 280.91 1.000

35 15.11 245.25 0.999

40 13.91 218.70 0.999

Glucose 5 39.43 1363.80 0.999

10 34.21 862.98 0.999

15 25.68 877.57 0.999

20 28.07 305.98 0.966

25 19.98 480.46 0.999

30 17.86 347.22 0.999

35 15.83 292.65 0.999

40 14.16 263.12 0.999

*)ηL = ηW + a(1-Aw)+ b(1-Aw)2

**)Speight(2005).

Sugar Temperature (℃) ηW a b R

Sucrose 5 61.68 5078.90 0.999

10 53.98 3605.50 1.000

15 45.76 2971.70 1.000

20 38.26 2529.00 1.000

25 33.85 2022.20 1.000

30 29.93 1673.60 0.998

35 26.43 1422.70 0.998

40 24.07 1192.60 0.997

Trehalose 5 46.18 7152.60 0.998

10 37.45 5520.70 0.999

15 35.00 4340.30 0.999

20 29.67 3626.80 0.999

25 26.52 2994.50 0.999

30 24.00 2486.30 0.999

35 21.90 2064.20 0.999

40 25.51 1335.20 0.999

Maltose 5 75.14 4735.60 0.999

10 50.75 4190.00 0.999

15 45.44 3305.40 0.999

20 38.90 2726.40 0.999

25 29.44 2562.40 0.999

30 26.85 2074.90 0.999

35 25.02 1654.00 0.999

40 25.65 1160.60 1.000

*)ηL=ηW + a(1-Aw)+ b(1-Aw)2

Table 1-1. Parameters in Eq. (2)* for viscosity of sugar solution as a function of water activity (monosaccharides).

Table 1-2. Parameters in Eq. (2)* for viscosity of sugar solution as a function of water activity (disaccharides).

-0.5

0

0.5

1

1.5

3.1 3.2 3.3 3.4 3.5 3.6

SucroseGlucoseRiboseWater

lnη

L(-)

1000/T(K-1)

at Aw = 0.985

Fig. 2. Arrhenius plot for viscosity of sugar solutions at Aw = 0.985.

Sugar Aw Ea(kJ/mol) R

No sugar added 1.000 17.5 0.997

Ribose 0.995 17.4 0.999

0.990 17.8 0.999

0.985 18.6 0.999

0.980 19.6 0.998

Mannose 0.995 18.0 0.998

0.990 18.8 0.998

0.985 19.8 0.998

0.980 20.8 0.998

Glucose 0.995 18.0 0.999

0.990 18.8 0.999

0.985 19.8 0.999

0.980 20.7 0.998

Sucrose 0.995 18.3 0.999

0.990 19.7 0.999

0.985 21.0 0.999

0.980 22.1 0.999

Trehalose 0.995 17.8 0.998

0.990 19.5 0.998

0.985 21.3 0.999

0.980 22.9 1.000

Maltose 0.995 18.6 0.998

0.990 20.1 0.998

0.985 21.3 0.999

0.980 22.3 1.000

Table 2. Activation energy for viscosity of sugar solutions.

235

In a sugar solution, there are three different intermolecu-

lar interactions: (1) water-water, (2) water-sugar, and (3)

sugar-sugar. Among these, the sugar-sugar interaction would

be weak for the following reason. For sugar solutions, Aw

was described by Eq. (1), from which the activity coefficient

of water, γW, is calculated to be exp(αXs2). As the parameter α is negative, as reported before (Miyawaki et al., 1997), γW

is less than unity, which suggests that the sugar-sugar inter-

action is negative. If the sugar-sugar interaction is positive, γW should be larger than unity (Kozak et al., 1968; Lilley,

1994). This means that the direct sugar-sugar interaction is

negligible in sugar solutions so that a slight increase in EA

with a decrease in Aw might be attributed to the sugar-water

interaction: hydration of sugar.

Intermolecular interaction among pectin molecules in so-

lutions without sugar Figure 3 shows the effects of temper-

ature and concentration on the viscosity of the citrus pectin

solution, ηH. Also in this case, Eq. (3) was applicable so that

the logarithm of ηH showed linear dependence on 1/T, and

the activation energy of viscosity, Ea, could be determined

from the slope. Ea increased from 17.5 to 31.9 kJ/mol with

an increase in pectin concentration up to 2.0%, as shown in

Figure 4. Kar and Arslan (1999b) reported Ea ranging from

19.6 to 27.2 kJ/mol for the viscosity of orange peel pectin

in concentrations ranging from 0.25 to 2.0%. Marcotte et al.

(2001) obtained Ea for viscosity of pectin ranging from 19.6

to 22.7 kJ/mol in pectin concentrations ranging from 1 to

5%.

Figure 4 also shows that the viscosity of pectin solution

increases exponentially with an increase in pectin concentra-

tion showing corresponding changes in the increase of pec-

tin-pectin interaction. With an increase in pectin concentra-

tion, the effect of water-pectin interaction increases, but this

effect is linearly proportional to the pectin concentration. The

pectin-pectin interaction, however, increases proportionally

to the square of pectin concentration so that an increase in Ea

with an increase in pectin concentration should be attributed

to the increase in the pectin-pectin interaction comprising

hydrogen-bonding and hydrophobic interaction for high-

methoxyl pectins (Oakenfull and Scott, 1984; Crandall and

Wicker, 1986; Oakenfull, 1991). Oakenfull and Scott (1984)

quantitatively evaluated the contributions of the hydrogen-

bonding and hydrophobic interactions to the pectin-pectin

interaction in the gelation process of high-methoxyl pectin.

They reported that the contributions of the hydrogen-bonding

and the hydrophobic interaction are -37.5 and -18.6 kJ/mol,

respectively, both of which are necessary to overcome the

entropy equivalent (+41.1 kJ/mol) for the configuration

change of the pectin molecule.

Intermolecular interaction among pectin molecules in

solutions with sugars Figure 5 shows the effect of the co-

existence of sugars on the viscosity, ηHL, of 2% citrus pectin

solutions. With an increase in sugar concentration, the vis-

cosity of pectin solution drastically increased. The extent

of viscosity increase is also dependent on the type of sugar.

Intermolecular Interaction of Pectin

-1

0

1

2

3

4

5

6

3.1 3.2 3.3 3.4 3.5 3.6

2.0%1.5%1%0.5%0%

lnη

H(-)

1000/T(K-1)

Fig. 3. Arrhenius plot for viscosity of citrus pectin solutions with concentration varied.

Fig. 4. Effect of concentration of citrus pectin on viscosity at 25℃ and activation energy for viscosity.

0

10

20

30

40

50

60

70

80

0 0.5 1 1.5 2

Ea(kJ/mol)η(mPa・s)

Ea(

kJ/mol),

ηH(mPa・

s)

Citrus pectin (%)

236

Sugar with stronger solvent-ordering activity was reported

to more greatly increase the viscosity (Sato and Miyawaki,

2000; Sato et al., 2004).

The dependence of ηHL on AW could be described by the

similar equation with Eq. (2) as follows:

ηHL = ηH + a’(1 – AW) + b’(1 – AW)2 (4)

where ηH, a’, and b’ are experimentally determined con-

stants. Although ηH should be a universal constant, theoreti-

cally, as was the case for ηW in Eq. (2), the actual value of ηH

was slightly different among experiments. Table 3 lists these

constants for 2% citrus pectin solutions with various sugars

at various concentrations with varied temperature, showing

good applicability of Eq. (4) with good correlation coeffi-

cients. For ηHL, Eq. (3) was also applicable so that the loga-

rithm of ηHL was plotted against 1/T in Figure 6 (AW = 0.985),

showing a good linear relationship. From the slope of this

plot, Ea was obtained and listed in Table 4 for 2% solutions

of citrus and apple pectin with various sugars at various AW.

From Table 4, Ea for viscosity of 2% citrus and apple

pectin solutions without sugar are 31.9 and 27.4 kJ/mol, re-

spectively. These values are much higher than that for pure

water (17.5 kJ/mol) because of the pectin-pectin interaction,

which has higher activation energy than water-water interac-

tion, as described previously. When sugars coexist, the acti-

vation energy for the viscosity of pectin solutions increased

more for an increase in sugar concentration. This increase

in Ea was dependent both on the AW and the type of sugars.

When compared at the same AW, Ea was much higher for di-

saccharide solutions than those of monosaccharide.

Y. Sato et al.

0

50

100

150

200

250

300

0.980.9850.990.9951

SucroseGlucoseRibose

ηHL(mPa・

s)

Aw

Fig. 5. Effect of coexistence of sugars on viscosity of citrus pectin solutions.

Sugar Temperature(℃) ηH

Ribose 5 213.84 10834 16121 0.999

10 166.03 5698 38350 0.999

15 125.52 4119 17313 0.997

20 101.62 2431 29134 0.993

25 75.16 3124 -31106 0.999

30 65.51 1798 -14161 0.980

35 55.72 1047 14671 0.990

40 45.61 1383 -16489 0.981

Mannose 5 230.07 16097 -33471 0.993

10 164.01 10293 27667 0.995

15 124.77 7333 -13837 0.993

20 98.79 4969 8520 0.995

25 78.63 4786 -60649 0.989

30 66.97 2568 5883 0.995

35 55.31 2381 -20072 0.984

40 49.00 918 47099 0.987

Glucose 5 218.35 11832 672060 0.993

10 154.98 12964 -26088 0.996

15 120.56 7172 94272 0.998

20 94.54 4989 100200 0.998

25 75.36 4665 -1343 0.995

30 61.79 4027 -20526 0.998

35 54.34 2212 21554 0.995

40 49.04 1310 35090 0.999

*)ηHL =ηH + a’(1-Aw) + b’(1-Aw)2

Rb’a’

Sugar Temperature (℃) ηH

Sucrose 5 210.00 53056 -2314800 1.000

10 150.00 36614 -1721500 1.000

15 124.08 9198 913700 0.993

20 92.68 11413 264830 0.990

25 73.43 8989 141430 0.994

30 67.10 6028 121910 0.994

35 61.64 3262 148300 0.988

40 55.32 3510 37613 0.953

Trehalose 10 152.70 26601 272980 1.000

15 119.84 15973 455290 1.000

20 103.24 -357 1617100 0.994

25 74.73 7347 520370 0.997

30 63.22 4946 429350 0.998

35 54.37 2747 373450 0.997

40 48.69 2694 223910 0.998

Maltose 5 196.70 53005 -2216000 1.000

10 161.84 19509 638370 0.997

15 124.67 5888 1275500 0.993

20 96.96 4835 831020 0.993

25 79.87 2751 675310 0.997

30 65.27 1938 509310 0.998

35 56.00 1468 387500 0.996

40 49.66 1123 274810 0.997

*)ηHL =ηH + a’(1-Aw) + b’(1-Aw)2

Rb’a’

Table 3-1. Parameters in Eq. (4)* for viscosity of 2% citrus pectin solutions with sugars as a function of water activity (monosaccha-rides).

Table 3-2. Parameters in Eq. (4)* for viscosity of 2% citrus pectin solutions with sugars as a function of water activity (disaccharides).

237

In the aqueous pectin solution with sugar, six intermo-

lecular interactions are expected to exist: (1) water-water,

(2) water-sugar, (3) sugar-sugar, (4) water-pectin, (5) sugar-

pectin, and (6) pectin-pectin. The activation energy for the

former two was small, as shown in Table 2, and the contribu-

tion of sugar-sugar interaction would not be as important as

previously discussed. Therefore, the latter three interactions

will be considered here.

In pectin solutions, the pectin-pectin interaction is more

important than water-pectin interaction when the pectin con-

centration is high as was described before. As for the effect

of sugars, Chen and Joslyn (1967) explained the viscosity-

enhancing property of sugar by the dehydration and the

hydrogen-bonding-formation activity that enhances the ag-

gregation among pectin molecules. Kar and Arsln (1999a)

attributed the viscosity-enhancing effect of sugars to the de-

crease in dielectric constant, dehydration action, and hydro-

gen bonding formation. Bulone et al. (2002) investigated the

role of sucrose in pectin gelation by static and dynamic light

scattering and reported that the apparent gyration radius of

the pectin molecule slightly increased with sucrose concen-

tration. Fishman et al. (2004) applied atomic force micros-

copy to observe nanostructure of native pectin sugar acid gel

and concluded that sugars strongly adsorb on pectin with the

ratio of bound sugar to pectin in excess of 100 to 1 (w/w).

We tried to measure the sugar-pectin interaction directly

by the measurement and comparison of AW of sugar solu-

tion with and without pectin. The AW values for 30% sucrose

solutions without and with 2% citrus pectin were 0.975 and

0.971, respectively. This insignificant difference between

the two strongly suggests that the direct interaction between

sugar and pectin is weak. In other cases, AW of sugar solution

should increase with the coexistence of pectin due to the de-

crease in the number of sugar molecules through the adsorp-

tion on pectin (Fishman et al., 2004) if a strong sugar-pectin

interaction exists (Kozak et al., 1968; Lilley, 1994).

Thus the rapid increase in viscosity of pectin solution

with the addition of sugars in Figure 5 is ascribed to the

increase in pectin-pectin interaction, which has higher acti-

vation energy than the other interactions. This explains the

increase in Ea for the viscosity of pectin solutions with an in-

crease in sugar concentration in Table 4. When sugars coex-

ist, sugars are strongly competitive in hydration with pectin

molecules so that the amount of hydration of pectin will be

reduced to increase the pectin-pectin interaction, which is re-

sponsible for the increase in Ea with a decrease in AW shown

in Table 4. This trend is, as shown in Figure 7, more obvious

for disaccharides than for monosaccharides because of the

higher solvent-ordering activity for the former.

As was reported by Oakenfull and Scott (1984), pectin-

pectin interaction comprises hydrogen bonding and hydro-

phobic interaction. For comparison at the same Aw, the ef-

fect of hydrogen bonding would be similar among different

sugars so that different effects on pectin-pectin interactions

among sugars should be responsible for the different effects

Intermolecular Interaction of Pectin

3.5

4

4.5

5

5.5

6

6.5

3.1 3.2 3.3 3.4 3.5 3.6

SucroseGlucoseRiboseWater

lnη

HL(-)

1000/T(K-1)

at Aw = 0.985

Fig. 6. Arrhenius plot for 2% citrus pectin solutions with various sugars at Aw = 0.985.

Sugar Citrus pectin Apple pectin

Aw Ea (kJ/mol) R Ea (kJ/mol) R

No sugar added 1.000 31.9 0.998 27.4 0.999

Ribose 0.995 33.8 0.997 29.8 0.998 0.990 35.5 0.996 31.7 0.999

0.985 37.0 0.996 32.4 0.999

0.980 38.5 0.996 31.3 0.999

Mannose 0.995 35.2 0.997 30.7 0.999

0.990 37.1 0.997 32.0 0.999

0.985 38.2 0.997 32.2 0.999

0.980 38.9 0.995 32.8 0.999

Glucose 0.995 34.2 0.997 30.8 0.997

0.990 37.1 0.997 32.2 0.999

0.985 36.8 0.999 31.9 1.000

0.980 37.2 0.997 32.0 0.999

Sucrose 0.995 35.9 0.990 35.8 0.992

0.990 36.2 0.998 33.2 0.999

0.985 41.4 0.998 33.2 0.999

0.980 45.5 0.995 36.1 1.000

Trehalose 0.995 35.7 0.992 35.7 0.999

0.990 36.9 0.999 32.1 0.998

0.985 44.5 0.995 33.4 0.999

0.980 41.1 1.000 34.5 1.000

Maltose 0.995 38.6 0.992 32.7 0.998

0.990 38.5 0.999 31.7 1.000

0.985 40.2 0.999 33.3 1.000

0.980 41.1 1.000 34.0 1.000

Table 4. Activation energy for viscosity for 2% pectin solutions with various sugars.

238

on the hydrophobic interaction through differences in the

solvent-ordering activity (Sato et al., 2004).

Acknowledgements We thank Ms. Takenaka, Ms. Koike, Ms. Hino,

and Ms. Takagi for their technical assistance. This study was partly

supported by a Grant-in-Aid for Scientific Research from Japan So-

ciety for the Promotion of Science (No. 14580144).

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Y. Sato et al.

30

35

40

45

-9-7-5-3-1

Citrus pectinApple pectin

Ea(kJ/mol)

Parameter α

R = 0.830

R = 0.931

at Aw = 0.985

Fig. 7. Correlation between the parameter α and the activa-tion energy for viscosity at Aw = 0.985.